Problem: $ -1.\overline{3} \div 0.\overline{91} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -13.3334...\\ x &= -1.3334...\end{align*} $ $\begin{align*} 9x &= -12 \\ x &= -\dfrac{12}{9}\end{align*} $ $\begin{align*} 100y &= 91.9191...\\ y &= 0.9191...\end{align*} $ $\begin{align*} 99y &= 91 \\ y &= \dfrac{91}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{12}{9} \div \dfrac{91}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{12}{9} \times \dfrac{99}{91} = {?} $ $ \phantom{-\dfrac{12}{9} \times \dfrac{91}{99}} = \dfrac{-12 \times 99}{9 \times 91} $ $ \phantom{-\dfrac{12}{9} \times \dfrac{91}{99}} = \dfrac{-12 \times \cancel{99}11} {\cancel{9} \times 91} $ $ \phantom{-\dfrac{12}{9} \times \dfrac{91}{99}} = -\dfrac{132}{91} $